Classical Electrodynamics · Problem 2.8

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · two-parallel-line-charges-capacitance

Problem

A two-dimensional potential problem is defined by two straight parallel line charges separated by a distance

R

with equal and opposite linear charge densities

\lambda

and

-\lambda

. (a) Show by direct construction that the surface of constant potential

V

is a circular cylinder (circle in the transverse dimensions) and find the coordinates of the axis of the cylinder and its radius in terms of

R

\lambda

, and

V

. (b) Use the results of part a to show that the capacitance per unit length

C

of two right-circular cylindrical conductors, with radii

a

and

b

, separated by a distance

d > a + b

, is

C = \frac{2\pi\epsilon_0}{\cosh^{-1}\!\left(\dfrac{d^2 - a^2 - b^2}{2ab}\right)}

C

agrees with the answer in Problem 1.7 in the appropriate limit and determine the next nonvanishing order correction in powers of

a/d

and

b/d

. (d) Repeat the calculation of the capacitance per unit length for two cylinders inside each other (

d < |b - a|

). Check the result for concentric cylinders (

d = 0